MFSR Chinook Salmon spawn timing analysis

1 Overview

Goal: Describe variation in MFSR Chinook Salmon spawn timing (i.e., putative redd completion date), and how it relates to environmental covariates.

Objectives:

1. compile stream reach-scale physical attributes and Chinook salmon spawning phenology from 2002 to 2005.

2. summarize variation in spawn timing across four years in eight streams.

3. fit a series of linear mixed-effects models to identify associations among environmental covariates and spawn timing.

1.1 Study Area

This study was conducted in the Middle Fork of the Salmon River (MFSR) in central Idaho (Fig. 1). The MFSR is a tributary of the Salmon River and is part of the larger Columbia River Basin. The MFSR is home to several species of salmon, including Chinook salmon (Oncorhynchus tshawytscha). Detailed study area information can be found in Thurow (2000), Minshall (1981), Servheen et al. (2001), and Thurow et al. (2020).

Figure 1: Map of the Middle Fork Salmon River watershed showing major tributaries and surveyed Chinook Salmon redd locations from 2002 to 2005.

2 Datasets

2.1 Spawn timing data

Spawn timing data for Chinook salmon were collected from 2001 to 2005 in the MFSR. Because redds were not observed daily, we inferred spawn dates as the initial date (day of year) a completed redd was observed.

We removed data from 2001, and data from Knapp Creek and Cape Horn Creek, as these sites were not consistently sampled.

We spatially joined each redd GPS location to the NHDPlus Version 2 (Horizon Systems, 2018) to assign stream reaches based on a common identifier (COMID). The COMID is used to link redd data with covariate data associated with the stream reach on which it is located. A summary of the dataset is shown in Table 1.

Table 1: Data summary

Name	spawn_data
Number of rows	3016
Number of columns	6
_______________________
Column type frequency:
Date	1
factor	4
numeric	1
________________________
Group variables	None

Variable type: Date

skim_variable	n_missing	complete_rate	min	max	median	n_unique
spawn_date	0	1	2002-08-02	2005-09-11	2003-08-22	115

Variable type: factor

skim_variable	n_missing	complete_rate	ordered	n_unique	top_counts
redd_id	0	1	FALSE	3016	129: 1, 129: 1, 129: 1, 129: 1
COMID	0	1	FALSE	104	235: 216, 235: 200, 235: 144, 235: 125
stream	0	1	FALSE	8	Big: 647, Bea: 479, Elk: 471, Mar: 368
year	0	1	FALSE	4	200: 1217, 200: 1187, 200: 404, 200: 208

Variable type: numeric

skim_variable	n_missing	complete_rate	mean	sd	p0	p25	p50	p75	p100
yday	0	1	239.7	8.36	206	234	241	246	263

2.1.1 Evaluate Grouping structure

The structure of the data is repeated measures on COMIDs, and multiple years, shared across COMIDs within streams.

• Redds observed at specific locations, COMID
• Nested within stream
• Observed across multiple years

Table 2 shows the number of COMIDs per stream, and Figure 2 shows the number of observations (unique redds) per COMID.

Many COMIDs only have 1-2 observation, so the groups are not well sampled. There are 23 COMIDs with <5 redds (26%), 13 with <= 2. (12.5%). With <5 obs/level, variance estimates can become unstable and lead to overfitting and absorbing noise (low AIC / high R2) and singular fits. Something to keep in mind during model fitting.

Table 2: Number of COMIDs per stream (major tributaries).

stream	n_COMIDs
Big	21
Loon	19
Camas	16
Marsh	13
Bear Valley	11
Sulphur	11
Beaver	7
Elk	6

Figure 2: Number of observations (redds) per COMID.

2.1.2 Spawn data descriptive statistics and visualizations

Figure 3 provides a visual summary of the distribution of spawn timing (day of year, yday), and variation by stream system and year.

Spawn timing is generally unimodal, with a peak in late August to early September. The distribution is slightly left skewed, but the mean and median are similar, indicating low skewness. The variance is low, suggesting no overdispersion. Suggests Poisson family for response if count of spawning events per day, or Gaussian if modeling day of year as continuous.

Variation by stream, year, and COMID (stream reach)

There is also substantial variation in spawn date by stream and year (Fig. 3B-C; Fig. 4) and within COMIDs within streams (Fig. 5).

Figure 3: Histogram and density of spawn timing for all streams and years (A), and by stream (B) and year (C). In panel (A), colored, vertical lines indicate the mean spawn date for each year., and the black vertical line indicated the global mean.

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Figure 4: Spawning phenology of adult Chinook Salmon. In all panels, the black density function represented stream-level spawn timing, while the colored density functions represent the spawn timing of individual years. The dashed vertical purple lines represent the 5th and 95th percentiles of the basin-wide spawn timing.

Figure 5: Spawn time variation by COMID and stream.

Cumulative proportion of redds over time

The temporal progression of Chinook salmon spawning activity varied within each stream across the four study years (Figure 6). The cumulative proportion of redds provides an intuitive measure of the spawning season’s pace and timing. Streams like Marsh and Sulphur exhibit rapid increases, suggesting short, concentrated spawning windows. In contrast, streams like Big and Camas show more gradual increases, indicating a more protracted spawning season. Year-to-year variation is evident in several streams. Overall, this figure highlights both spatial and temporal heterogeneity in spawn timing that underpins the need for models incorporating stream- and year-specific effects.

Figure 6: Proportion of cumulative Chinook salmon redds over time (day of year) across years (2002–2005) and streams. Each line represents a different year, with color denoting the year. Stream-specific panels illustrate temporal variation in the progression of spawning activity, as measured by cumulative redd counts normalized to the maximum value in each stream-year combination.

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2.2 Covariate Derivation and Screening

To identify environmental predictors of Chinook salmon spawn timing, we quantified covariates that describe thermal and physical stream conditions at redd locations. Covariates were selected based on three criteria: (1) demonstrated influence on salmon phenology in prior literature, (2) availability at all redd locations (i.e., COMIDs), and (3) low collinearity with other predictors.

The initial focal covariates included stream temperature (°C), stream discharge (cms), elevation (m), and stream slope (unitless). Elevation and slope were extracted from NHDPlus (Moore et al. 2019).

2.2.1 Elevation and slope

Elevation and stream slope data were available at the COMID (stream reach) scale from the NHDPlus Version 2 (Horizon Systems, 2018).

2.2.2 Stream temperature

We used modeled daily average stream temperature values predicted at the stream segment (COMID) scale (Siegel et al. 2023). These data were downloaded and filtered to 2002-2005 and for the MFSR. Figure 7 shows the modeled thermal regimes for MFSR tributaries.

Figure 7: Modeled thermal regimes (2001-2005) for MFSR tributaries.

These modeled temperatures were validated against empirical logger data collected at a subset of sites in the Middle Fork basin, with strong agreement (R² > 0.9).

ADD PLOT OF COMPARISON BETWEEN EMPIRICAL AND MODELED TEMPPS

2.2.3 Discharge (streamflow)

Stream flow data were compiled from a single USGS Gage lower in the watershed (MF Salmon River at MF Lodge NR Yellow Pine ID - 13309220; Figure 8).

Figure 8: Inter-annual variability in daily discharge (cfs) at MF Lodge USGS Gage 13309220.

2.2.4 Time-windowed temp and flow metrics

For each redd, we computed several time-windowed summaries of temperature and flow relative to the inferred spawn date:

• Antecedent metrics: average conditions over 30, 60, and 90 days preceding the spawn date (e.g., temp_90_before, flow_60_before),
• Spanning metrics: averages over time windows centered on the spawn date,
• Post-spawn metrics: 30-, 60-, and 90-day averages following the spawn date,
• Time-invariant metrics: summaries calculated relative to a fixed calendar date (e.g., August 1) representing an approximate onset of the spawning window.

The time invariant and after metrics were omitted from further consideration as preliminary data exploration showed weak if any relationship with spawn timing.

3 Exploratory Data Analysis

3.1 Bivariate relationships

We did exploratory data analysis on the compiled dataset to identify candidate covariates for inclusion in model selection.

Bivariate relationships between spawn date (yday) and continuous predictors (Figure 9), and pairwise correlations between all continuous covariates (Figure 10), are shown below.

Figure 9: Bivariate relationships between spawn date (yday) and continuous covariates. Solid lines are linear fits.

Figure 10: Pairwise correlations between continuous covariates.

Takeaways from Figures 9 and 10:

Amoung temperature variables, temp_90 clearly shows the strongest relationship with spawn date (Figure 9), and is highly colinear with temp_30 and temp_60 (Figure 10).

The is a weak negative relationship between mean_elevation and yday (Figure 9), and it is weakly correlated with temp_90 (0.31, Figure 10).

• flow_30 = decaying exponential, obvious year effect
• flow_60 = ditto
• flow_90 = inflections, year effect clear
• slope = no relationship

3.2 temp_90

Figure 11 shows the relationship between temp_90 and spawn date by stream and year.

Figure 11: Relationship between temp_90 and yday (spawn date) by stream and year. Solid lines are linear fits.

We evaluated the role of temp_90 in explaining variation in Chinook salmon spawn timing:

m.t90.1 <- lmer(yday ~ (1|COMID), data = df_mod, REML = FALSE)
m.t90.2 <- lmer(yday ~ temp_90 + (1|COMID), data = df_mod, REML = FALSE)
m.t90.3 <- lmer(yday ~ temp_90 + I(temp_90^2) + (1|COMID), data = df_mod, REML = FALSE)

The model with a quadratic effect of temp_90 (m.t90.3) improved dramatically over the null (intercept-only) model and linear effect (Table 3), and boosting both conditional and marginal R² (Table 4). This confirms a strong, nonlinear effect of pre-spawn temperature on spawn timing (Figure 12).

Table 3: AIC comparison of linear mixed models relating spawn date (yday) to temp_90.

Model	df	AIC	delta
m.t90.3	5	15848.81	0.0000
m.t90.2	4	16570.12	721.3107
m.t90.1	3	18629.30	2780.4987

Table 4: Model performance of linear mixed models relating spawn date (yday) to temp_90.

Name	Model	R2_conditional	R2_marginal	ICC	RMSE	AIC_wt	BIC_wt	Performance_Score
m.t90.3	lmerMod	0.9177407	0.7643117	0.6509825	3.108315	1	1	0.8333333
m.t90.2	lmerMod	0.9086448	0.6915320	0.7038423	3.489863	0	0	0.6100606
m.t90.1	lmerMod	0.6571743	0.0000000	0.6571743	4.929494	0	0	0.0195227

Figure 12: Predicted marginal effect of temp_90 on yday (spawn date) for (A) m.t90.2 and (B) m.t90.3.

Next, we add in stream and year as main effects, and then interactions between temp_90 and stream and year:

m.t90.4 <- lmer(yday ~ temp_90 + I(temp_90^2) + stream + (1|COMID), data = df_mod, REML = FALSE)
m.t90.5 <- lmer(yday ~ temp_90 + I(temp_90^2) + year + (1|COMID), data = df_mod, REML = FALSE)
m.t90.6 <- lmer(yday ~ temp_90 + I(temp_90^2) + stream + year + (1|COMID), data = df_mod, REML = FALSE)
m.t90.7 <- lmer(yday ~ temp_90 * stream + I(temp_90^2) + year + (1|COMID), data = df_mod, REML = FALSE)
m.t90.8 <- lmer(yday ~ temp_90 * year + I(temp_90^2) + stream + (1|COMID), data = df_mod, REML = FALSE)

Table 5 shows that adding stream (m.t90.4) or year (m.t90.5) individually improved model fit relative to m.t90.3, but year alone (m.t90.5) performed better than stream alone (m.t90.4).

Combining stream and year additively (m.t90.6) provided further improvement in AIC, along with the best marginal R2 and RMSE (Table 6). This suggests that both stream and year provide meaningful structure in explaining variation in spawn timing.

Adding the interaction between temperature and stream (m.t90.7) increased complexity, slightly lowered the marginal R2, and has high multicollinearity (VIF ~15).

Table 5: AIC comparison of linear mixed models relating spawn date (yday) to temp_90.

Model	df	AIC	delta
m.t90.8	18	12136.83	0.0000
m.t90.7	22	13038.01	901.1783
m.t90.6	15	13473.12	1336.2837
m.t90.5	8	13531.17	1394.3321
m.t90.4	12	15807.98	3671.1454
m.t90.3	5	15848.81	3711.9726

Table 6: Model performance of linear mixed models relating spawn date (yday) to temp_90.

Name	Model	R2_conditional	R2_marginal	ICC	RMSE	AIC_wt	BIC_wt	Performance_Score
m.t90.8	lmerMod	0.9928189	0.6448836	0.9797782	1.581472	1	1	0.8333333
m.t90.7	lmerMod	0.9852284	0.7177821	0.9476590	1.865263	0	0	0.5267941
m.t90.6	lmerMod	0.9800601	0.7293886	0.9263154	2.022117	0	0	0.5061439
m.t90.5	lmerMod	0.9871140	0.6600787	0.9620912	2.021778	0	0	0.4528690
m.t90.3	lmerMod	0.9177407	0.7643117	0.6509825	3.108315	0	0	0.2196386
m.t90.4	lmerMod	0.8982595	0.7921275	0.5105629	3.110542	0	0	0.1666667

Adding the interaction between temperature and year (m.t90.8) gave the lowest AIC overall but had the lowest marginal R² of all models tested. This means that while the model “fit” the data better on paper (AIC), it did so by overfitting or by capturing year-to-year idiosyncrasies that may not generalize well to new data (suspicious marginal effect, Figure 13).

Figure 13: Predicted marginal effect (m.t90.8) of temp_90 on yday (accounting for the effects of stream and year).

The suspicious curvature in the marginal effect plot of m.t90.8 (showing an unrealistic predicted relationship between temp_90 and yday) confirms the risk of confounding or overfitting with interaction terms—particularly given that the random intercepts already capture substantial site-level variation.

Given the trade-offs, model m.t90.6 emerged as the best compromise: it retained additive main effects of temperature, stream, and year, had high conditional and marginal R², moderate AIC, low RMSE, and avoided high multicollinearity.

3.2.1 Summary

The exploratory analysis of temperature effects on spawn timing revealed a clear, nonlinear relationship between pre-spawn temperature and the day of year when spawning occurs. A quadratic term for temperature consistently improved model fit over linear-only specifications. When adding stream and year as main effects, model performance improved substantially, with both factors accounting for additional variation in spawn timing. However, introducing interaction terms between temperature and either stream or year led to either inflated multicollinearity (VIFs > 10) or implausible model predictions, indicating potential overfitting. Although the temperature-by-year interaction model (m.t90.8) had the lowest AIC, it yielded a counterintuitive relationship between temperature and spawn timing. Therefore, the additive model with quadratic temperature and main effects of stream and year (m.t90.6) was selected as the preferred model, balancing goodness-of-fit with interpretability and biological realism.

3.3 mean_elevation

Figure 14 shows the relationship between between (A) mean_elevation and temp_90 by stream and (B) mean_elevation and yday (spawn date) by stream and year.

Figure 14: Relationship between (A) mean_elevation and temp_90 by stream and (B) mean_elevation and yday (spawn date) by stream and year. Solid lines are linear fits.

We evaluated the role of mean elevation in explaining variation in Chinook salmon spawn timing:

m.ele.1 <- lmer(yday ~ (1|COMID), data = df_mod, REML = FALSE)
m.ele.2 <- lmer(yday ~ mean_elevation + (1|COMID), data = df_mod, REML = FALSE)
m.ele.3 <- lmer(yday ~ mean_elevation + stream + (1|COMID), data = df_mod, REML = FALSE)
m.ele.4 <- lmer(yday ~ mean_elevation + year + (1|COMID), data = df_mod, REML = FALSE)
m.ele.5 <- lmer(yday ~ mean_elevation + stream + year + (1|COMID), data = df_mod, REML = FALSE)
m.ele.6 <- lmer(yday ~ mean_elevation * stream + (1|COMID), data = df_mod, REML = FALSE)
m.ele.7 <- lmer(yday ~ mean_elevation * year + (1|COMID), data = df_mod, REML = FALSE)

Including mean elevation as a main effect improved model fit (ΔAIC = 19 between m.ele.1 and m.ele.2). The best-supported model (m.ele.5) included mean elevation alongside stream and year (ΔAIC = 281 between m.ele.2 and m.ele.5), with moderate collinearity (VIFs ~6), suggesting that mean elevation can contribute to explaining spawn timing when controlling for stream and year (15).

However, adding interactions with stream or year substantially increased model complexity and collinearity, as evidenced by extremely high VIFs (>45). Given the known inverse relationship between elevation and temperature across streams (Figure 14A), caution is warranted when later incorporating both elevation and temperature in the same model.

Table 7: AIC comparison of linear mixed models relating spawn date (yday) to mean_elevation.

Model	df	AIC	delta
m.ele.5	14	18329.29	0.00000
m.ele.6	18	18371.62	42.33148
m.ele.3	11	18431.88	102.58771
m.ele.7	10	18435.12	105.82567
m.ele.4	7	18501.44	172.14764
m.ele.2	4	18611.27	281.97710
m.ele.1	3	18629.30	300.01337

Table 8: Model performance of linear mixed models relating spawn date (yday) to mean_elevation.

Name	Model	R2_conditional	R2_marginal	ICC	RMSE	Sigma	AIC_wt	AICc_wt	BIC_wt	Performance_Score
m.ele.5	lmerMod	0.6399737	0.5939968	0.1132428	4.876512	4.927506	1	1	1	0.6843974
m.ele.7	lmerMod	0.6684326	0.1243275	0.6213568	4.774852	4.851946	0	0	0	0.5175646
m.ele.4	lmerMod	0.6574860	0.1155940	0.6127186	4.835441	4.913296	0	0	0	0.4006749
m.ele.1	lmerMod	0.6571743	0.0000000	0.6571743	4.929494	5.009972	0	0	0	0.2596896
m.ele.2	lmerMod	0.6503692	0.1014398	0.6108988	4.929830	5.009158	0	0	0	0.2505245
m.ele.6	lmerMod	0.6398941	0.6301923	0.0262347	4.989054	5.016809	0	0	0	0.1678027
m.ele.3	lmerMod	0.6269782	0.5838335	0.1036716	4.971609	5.022058	0	0	0	0.1413263

Table 9: Variance inflation factors for model m.ele.5.

	GVIF	Df	GVIF^(1/(2*Df))
mean_elevation	5.739424	1	2.395709
stream	5.943076	7	1.135760
year	1.042077	3	1.006893

Table 9: Variance inflation factors for model m.ele.6.

	GVIF	Df	GVIF^(1/(2*Df))
mean_elevation	2.317261e+03	1	48.137937
stream	9.622172e+10	7	6.088629
mean_elevation:stream	5.226657e+11	7	6.870938

Figure 15: Estimated relationship between spawn date and mean elevation (m.ele.5).

3.4 slope

slope is not related to yday (Figure 16A), though there is some association between slope and yday when considering stream (Figure 16B). We will likely drop slope.

Figure 16: Bivariate relationship between slope and spawn date (A) and by stream (B). Solid lines are linear fits.

3.5 flow_90

Because flow data are not COMID- or stream-specific, it makes sense to think about and represent flow as an out-of-basin year effect that determines when adults make it back to the MFSR and initially onto the spawning grounds.

The strong correlation between flow_90 and year can be seen clearly in (Figure 17A).

Figure 17: Spawn timing vs. 90-day mean discharge pre spawn at MF Lodge.

Next we compare the following simple linear models to examine functional structure:

# flow_90 models
m1 <- lm(yday ~ flow_90, data = model_data)
m2 <- lm(yday ~ I(flow_90^2), data = model_data)
m3 <- lm(yday ~ flow_90 + year, data = model_data)
m4 <- lm(yday ~ flow_90 + year + stream, data = model_data)
m5 <- lm(yday ~ flow_90 * stream, data = model_data)
m6 <- lm(yday ~ flow_90 * stream + year, data = model_data)
m7 <- lm(yday ~ flow_90 * year + stream, data = model_data)
m8 <- lm(yday ~ flow_90 * stream + I(flow_90^2), data = model_data)
m9 <- lm(yday ~ flow_90 * stream + year + I(flow_90^2), data = model_data)

AIC scores suggest m7 is best model (Table 10). However, while the R^2 value is 0.98, the parameter estimate for flow_90 has is incredibly small and most variation is now attributed to the year contrasts (Table 11). Thus, flow_90 is clearly confounded with year, and confirmed by high Variance Inflation Factor (VIF) scores (Figure 18).

Table 10: Model performance of linear models relating spawn date (yday) to flow_90.

Name	Model	R2	R2_adjusted	RMSE	Sigma	AIC_wt	AICc_wt	BIC_wt	Performance_Score
m7	lm	0.9780529	0.9779505	1.237710	1.240800	1	1	1	1.0000000
m9	lm	0.9407030	0.9403269	2.034449	2.041229	0	0	0	0.4960534
m6	lm	0.9108990	0.9103638	2.493858	2.501751	0	0	0	0.4472891
m4	lm	0.8913935	0.8909958	2.753330	2.758824	0	0	0	0.4181935
m3	lm	0.8745347	0.8743681	2.959322	2.961778	0	0	0	0.3942842
m8	lm	0.7348812	0.7334668	4.301804	4.313980	0	0	0	0.2174139
m5	lm	0.7130109	0.7115760	4.475722	4.487641	0	0	0	0.1921569
m1	lm	0.5900974	0.5899614	5.348977	5.350752	0	0	0	0.0576227
m2	lm	0.5352432	0.5350890	5.695650	5.697539	0	0	0	0.0000000

Table 11: Parameter estimates for m7, yday ~ flow_90 * year + stream.

Parameter	Coefficient	SE	CI	CI_low	CI_high	t	df_error	p
(Intercept)	278.2566486	0.2130585	0.95	277.8388932	278.6744041	1306.0106434	3001	0.0000000
flow_90	-0.0219390	0.0001136	0.95	-0.0221617	-0.0217163	-193.1695196	3001	0.0000000
year2003	-8.3607844	0.2371172	0.95	-8.8257131	-7.8958557	-35.2601350	3001	0.0000000
year2004	11.0240656	0.3856186	0.95	10.2679620	11.7801692	28.5879998	3001	0.0000000
year2005	1.0123700	0.7619753	0.95	-0.4816766	2.5064167	1.3286127	3001	0.1840768
streamBeaver	0.5591186	0.1048594	0.95	0.3535151	0.7647220	5.3320808	3001	0.0000001
streamBig	-0.4395848	0.0772852	0.95	-0.5911220	-0.2880475	-5.6878290	3001	0.0000000
streamCamas	-0.0934454	0.0899470	0.95	-0.2698094	0.0829186	-1.0388944	3001	0.2989375
streamElk	0.2219142	0.0861007	0.95	0.0530918	0.3907365	2.5773800	3001	0.0100026
streamLoon	0.0895360	0.1073905	0.95	-0.1210304	0.3001024	0.8337428	3001	0.4044923
streamMarsh	-0.7143977	0.0913025	0.95	-0.8934196	-0.5353759	-7.8245141	3001	0.0000000
streamSulphur	-0.0665471	0.1017872	0.95	-0.2661269	0.1330327	-0.6537861	3001	0.5132997
flow_90:year2003	0.0094506	0.0001190	0.95	0.0092172	0.0096840	79.3900831	3001	0.0000000
flow_90:year2004	-0.0101387	0.0002436	0.95	-0.0106163	-0.0096610	-41.6197601	3001	0.0000000
flow_90:year2005	-0.0046105	0.0005766	0.95	-0.0057411	-0.0034799	-7.9957348	3001	0.0000000

Figure 18: Variance inflation factors (VIF) for model m7, yday ~ flow_90 * year + stream.

3.5.1 Why flow_90 is problematic

Not spatially resolved

• We are modeling spawn timing at the redd level (COMID/stream)
• But flow_90 is calculated from a single downstream gauge, and applied to all redds
• This assumes flow conditions are identical across all sites
• Including it gives the illusion of spatially resolved variation that isn’t there

Correlated with year

• Since flow_90 varies mostly across years, it is strongly confounded with year
• Any flow-related signal is probably already captured by your year fixed effect
• Including both flow_90 and year risks collinearity, and may produce misleading inferences

Spawn-time aligned flow ≠ experienced flow

• While flow_90 is aligned to each redd’s spawn date, it still reflects a lower-basin gauge, not the actual hydrologic conditions experienced at the redd site
• So it might be precisely wrong — aligned in time but irrelevant in space

Recommendation:

Drop flow_90 from model.

Although we initially considered including 90-day mean streamflow (flow_90) as a predictor of spawn timing, this variable was ultimately excluded due to concerns about ecological validity and model overfitting. Stream flow data were derived from a single downstream USGS gauge and did not capture spatial variation across the study streams or reaches. Moreover, because flow_90 was closely aligned with year, it introduced strong collinearity with the year effect and risked attributing site-level variation to flow patterns not actually experienced by individual redds. As such, we excluded flow_90 to avoid misleading inference.

4 Model fitting

4.1 Final dataset

The final dataset includes:

• yday: spawn date, continuous response variable
• comid, stream, year: grouping variables
• temp_90: 90-day mean temperature pre-spawn, continuous predictor variable
• slope and mean_elevation: provisionally retaining continuous predictor variable

We scaled the continuous covariates to have a mean of 0 and standard deviation of 1. This is important for mixed models, as it helps with convergence and interpretation (Table 12).

Table 12: Final dataset for modeling, first 5 rows.

yday	COMID	stream	year	temp_90	mean_elevation	slope
235	23519365	Bear Valley	2002	-0.3178564	0.7181159	-0.5189141
235	23519365	Bear Valley	2002	-0.3178564	0.7181159	-0.5189141
235	23519319	Bear Valley	2002	0.4665480	0.6954808	-0.7107062
235	23519319	Bear Valley	2002	0.4665480	0.6954808	-0.7107062
235	23519319	Bear Valley	2002	0.4665480	0.6954808	-0.7107062

4.2 Model specification

Variable	Fixed?	Random?	Why
COMID	No	Yes	Not estimating COMID effects, just accounting for correlation, repeated measures
Stream	Yes	Maybe	8 streams to compare
Year	Yes	Maybe	4 years to compare

Here, COMID will be included in all models as a random (intercept) effect to account for repeated measures on COMIDs. Stream will be treated as a fixed effect to compare average effects across streams, and year as a fixed effect to compare average effects across years. The later maybe be better included as a random effect, as it is not a treatment effect, but rather a random sample of years, but we only have 4 years of data, so we will treat it as a fixed effect for now.

4.3 Model comparison

We used linear mixed-effects models to evaluate environmental predictors of Chinook salmon spawn timing, with redd observation day-of-year (yday) as the response variable. First, we scaled the continuous covariates to have a mean of 0 and standard deviation of 1 to assist convergence and interpretation.

We fit 31 additive models representing all combinations of fixed effects: temp_90, stream, year, slope, and mean_elevation. All models included a random intercept for COMID to account for repeated measures across stream reaches. Further, based on exploratory analysis and biological expectations of nonlinear thermal responses, temperature effects were modeled using both linear and quadratic terms for the 90-day average stream temperature prior to spawning (temp_90 and I(temp_90^2)). Model comparison was performed using AIC, and model fit was evaluated using marginal and conditional R², RMSE, and intraclass correlation coefficients (ICC). All models were fit using maximum likelihood (REML = FALSE) to enable consistent comparison of differing fixed effects.

m1 <- lmer(yday ~ temp_90 + I(temp_90^2) + (1|COMID), data = df_mod, REML = FALSE)
m2 <- lmer(yday ~ stream + (1|COMID), data = df_mod, REML = FALSE)
m3 <- lmer(yday ~ year + (1|COMID), data = df_mod, REML = FALSE)
m4 <- lmer(yday ~ mean_elevation + (1|COMID), data = df_mod, REML = FALSE)
m5 <- lmer(yday ~ slope + (1|COMID), data = df_mod, REML = FALSE)

m6  <- lmer(yday ~ temp_90 + I(temp_90^2) + stream + (1|COMID), data = df_mod, REML = FALSE)
m7  <- lmer(yday ~ temp_90 + I(temp_90^2) + year + (1|COMID), data = df_mod, REML = FALSE)
m8  <- lmer(yday ~ temp_90 + I(temp_90^2) + mean_elevation + (1|COMID), data = df_mod, REML = FALSE)
m9  <- lmer(yday ~ temp_90 + I(temp_90^2) + slope + (1|COMID), data = df_mod, REML = FALSE)
m10 <- lmer(yday ~ stream + year + (1|COMID), data = df_mod, REML = FALSE)
m11 <- lmer(yday ~ stream + mean_elevation + (1|COMID), data = df_mod, REML = FALSE)
m12 <- lmer(yday ~ stream + slope + (1|COMID), data = df_mod, REML = FALSE)
m13 <- lmer(yday ~ year + mean_elevation + (1|COMID), data = df_mod, REML = FALSE)
m14 <- lmer(yday ~ year + slope + (1|COMID), data = df_mod, REML = FALSE)
m15 <- lmer(yday ~ mean_elevation + slope + (1|COMID), data = df_mod, REML = FALSE)

m16 <- lmer(yday ~ temp_90 + I(temp_90^2) + stream + year + (1|COMID), data = df_mod, REML = FALSE)
m17 <- lmer(yday ~ temp_90 + I(temp_90^2) + stream + mean_elevation + (1|COMID), data = df_mod, REML = FALSE)
m18 <- lmer(yday ~ temp_90 + I(temp_90^2) + stream + slope + (1|COMID), data = df_mod, REML = FALSE)
m19 <- lmer(yday ~ temp_90 + I(temp_90^2) + year + mean_elevation + (1|COMID), data = df_mod, REML = FALSE)
m20 <- lmer(yday ~ temp_90 + I(temp_90^2) + year + slope + (1|COMID), data = df_mod, REML = FALSE)
m21 <- lmer(yday ~ temp_90 + I(temp_90^2) + mean_elevation + slope + (1|COMID), data = df_mod, REML = FALSE)
m22 <- lmer(yday ~ stream + year + mean_elevation + (1|COMID), data = df_mod, REML = FALSE)
m23 <- lmer(yday ~ stream + year + slope + (1|COMID), data = df_mod, REML = FALSE)
m24 <- lmer(yday ~ stream + mean_elevation + slope + (1|COMID), data = df_mod, REML = FALSE)
m25 <- lmer(yday ~ year + mean_elevation + slope + (1|COMID), data = df_mod, REML = FALSE)

m26 <- lmer(yday ~ temp_90 + I(temp_90^2) + stream + year + mean_elevation + (1|COMID), data = df_mod, REML = FALSE)
m27 <- lmer(yday ~ temp_90 + I(temp_90^2) + stream + year + slope + (1|COMID), data = df_mod, REML = FALSE)
m28 <- lmer(yday ~ temp_90 + I(temp_90^2) + stream + mean_elevation + slope + (1|COMID), data = df_mod, REML = FALSE)
m29 <- lmer(yday ~ temp_90 + I(temp_90^2) + year + mean_elevation + slope + (1|COMID), data = df_mod, REML = FALSE)
m30 <- lmer(yday ~ stream + I(temp_90^2) + year + mean_elevation + slope + (1|COMID), data = df_mod, REML = FALSE)

m31 <- lmer(yday ~ temp_90 + I(temp_90^2) + stream + year + mean_elevation + slope + (1|COMID), data = df_mod, REML = FALSE)

m26a <- lmer(yday ~ temp_90 + I(temp_90^2) + year  + stream * mean_elevation + (1|COMID), data = df_mod, REML = FALSE)

Table 13: AIC selection performance metrics for additive linear models; top 10 of 31 shown.

Model	df	AIC	delta_AIC	R2_marginal	R2_conditional	RMSE	ICC
m26	16	13356.04	0.000	0.785	0.956	2.023	0.796
m31	17	13358.04	1.998	0.785	0.956	2.023	0.795
m27	16	13468.96	112.925	0.737	0.979	2.022	0.922
m16	15	13473.12	117.079	0.729	0.980	2.022	0.926
m29	10	13487.82	131.783	0.727	0.984	2.022	0.942
m19	9	13488.73	132.690	0.726	0.984	2.022	0.943
m7	8	13531.17	175.128	0.660	0.987	2.022	0.962
m20	9	13532.90	176.858	0.660	0.987	2.022	0.962
m17	13	15787.25	2431.210	0.783	0.880	3.112	0.446
m28	14	15789.04	2433.003	0.783	0.880	3.112	0.445

Based on AIC, the best-fitting model was m26, though its improvement in AIC and performance over m31, the full model which includes slope, was modest (Table 13 and ??). Suggests little benefit to including slope as a predictor.

m27, which excludes mean_elevation, had substantially worse AIC and performance than m26 and m31, indicating that elevation contributes meaningfully to explaining spawn timing variation.

Model m16, which excludes both mean_elevation and slope, had nearly identical predictive accuracy (RMSE) and higher conditional R² compared to m26 and m31, despite a slightly lower (0.73 vs 0.78) marginal R². The 117-point AIC difference between m16 and m26 likely reflects subtle improvements in likelihood fit due to topographic covariates.

Visual inspection of partial (model-based) relationships (Figure 19 B)) revealed that the modeled elevation effect was inconsistent with ecological expectations.

In this case, m26 is picking up a positive association: after controlling for temperature and stream identity, higher-elevation reaches are predicted to spawn later. This is contrary to the observed negative relationships between elevation, temperature, and spawn timing in several streams (Figure 14).

Figure 19: Predicted relationship (red line) between spawn date (yday) and (A) temp_90, (B) mean_elevation.

This could reflect either (a) a real but subtle ecological effect, or (b) statistical artifact from collinearity/confounding.

1. There are some reasons a residual positive elevation effect might make sense:

• Hydrology and snowmelt: At higher elevations, cooler conditions might delay snowmelt-driven flow cues, potentially pushing back spawning independent of temperature.

• Population differences: If higher-elevation reaches host subpopulations adapted to shorter growing seasons, they might time spawning later to avoid fry emerging into very harsh winter conditions.

• Migration lags: Reaches further upstream (which often correspond to higher elevations) could simply take longer for fish to reach, even if temperatures are similar.

But these are second-order hypotheses—we’d need strong ecological justification to lean on them.

2. Statistical red flags

• The positive effect could just as easily be an artifact of collinearity. Elevation and temperature are tightly coupled in your dataset. Once you partial out temperature, the remaining variation in elevation is limited and unstable.

• The fact that predicted vs. observed plots showed inconsistent elevation effects across streams (your Figure elev-plots) suggests this may not be a robust, generalizable signal.

Variance inflation factors (VIFs) from m26 indicated moderate collinearity between elevation and stream (VIF = 6.50 and 6.48, respectively). Elevation was strongly confounded with stream identity in our dataset (Figure 14), with high-elevation streams showing considerable overlap in spawn timing and temperature.

##                    GVIF Df GVIF^(1/(2*Df))
## temp_90        1.786743  1        1.336691
## I(temp_90^2)   1.323720  1        1.150530
## stream         6.496841  7        1.143010
## year           1.814029  3        1.104352
## mean_elevation 6.477001  1        2.544995

I suggest we acknowledge both interpretations:

• Statistically: The positive elevation effect emerges after controlling for temperature, but it is likely confounded by collinearity.

• Ecologically: A true positive effect could be consistent with delayed migration, snowmelt-driven cues, or subpopulation adaptations, but our dataset can’t fully disentangle these mechanisms.

• Conclusion: Because the elevation effect runs counter to raw data patterns and shows signs of confounding, we made the conservative decision to omit it from the final model set.

4.4 Targeted model comparison

We compared our top additive model (m16; fixed effects: temp_90, I(temp_90^2), stream, year; random intercept for COMID) to a series of increasingly flexible models to evaluate the contribution of random slopes and temperature nonlinearity.

4.4.1 Random slopes for temp_90

We then tested whether model performance improved by allowing the effect of temperature to vary across stream reaches by adding a random slope for temp_90 to the COMID grouping factor. This model was fit using restricted maximum likelihood (REML), and its fit was compared to the simpler random intercept model.

To account for potential variation in temperature sensitivity across sites, we extended m16 to include COMID-specific random slopes for temp_90, yielding a model with the same fixed effects but with the random structure (1 + temp_90 | COMID). Because the random effects structure differed, we fit both models using restricted maximum likelihood (REML).

The random slope model (m16_rs) substantially improved fit (ΔAIC = 510), reduced RMSE (2.02 → 1.78 days), and slightly increased conditional R² (0.981 → 0.985). However, the marginal R² declined from 0.714 to 0.698, reflecting a redistribution of explanatory power from fixed to random effects. This shift is expected when site-specific variation in temperature responses is modeled explicitly, and suggests that temperature sensitivity varies meaningfully among stream reaches.

m16 <- lmer(
  yday ~ temp_90 + I(temp_90^2) + stream + year + (1 | COMID),
  data = df_mod, REML = TRUE
  )
m16_rs <- lmer(
  yday ~ temp_90 + I(temp_90^2) + stream + year + (1 + temp_90 | COMID), 
  data = df_mod, REML = TRUE
  )

Table 14: Model selection for m16 vs. m16_rs.

Model	df	AIC	delta_AIC	R2_marginal	R2_conditional	RMSE	ICC
m16_rs	17	12948.76	0.000	0.698	0.985	1.781	0.949
m16	15	13459.05	510.283	0.714	0.981	2.022	0.932

4.4.2 Partitioning Variation Between Nonlinearity and Heterogeneity

To determine whether the quadratic temperature term remained necessary in the presence of random slopes, we re-fit the random slope model with and without I(temp_90^2) using maximum likelihood (ML).

The full model including both quadratic temperature and random slopes (m16_rs) outperformed the linear version (m16_rs_noquad) by ΔAIC = 20.4, despite only one additional degree of freedom. RMSE and R² values were nearly identical, but the full model retained a slightly better fit, particularly at the tails of the temperature distribution. This indicates that while random slopes capture most of the temperature-related variation, the quadratic term meaningfully refines the relationship without overfitting or destabilizing the model.

m16_rs <- lmer(
  yday ~ temp_90 + I(temp_90^2) + stream + year + (1 + temp_90 | COMID), 
  data = df_mod, REML = FALSE
  )
m16_rs_noquad <- lmer(
  yday ~ temp_90 + stream + year + (1 + temp_90 | COMID),
  data = df_mod, REML = FALSE
  )

Table 15: Model selection for m16_rs vs. m16_rs_noquad.

Model	df	AIC	delta_AIC	R2_marginal	R2_conditional	RMSE	ICC
m16_rs	17	12965.97	0.000	0.713	0.984	1.782	0.945
m16_rs_noquad	16	12986.41	20.441	0.714	0.984	1.788	0.943

Figure 20: Model predictions (red lines) for m16_rs and m16_rs_noquad.

4.4.3 Summary of random slopes and nonlinearity

For results

To account for site-level variation in thermal sensitivity, we extended this model by allowing COMID-specific random slopes for temperature. This significantly improved model fit (ΔAIC = 510), reduced prediction error (RMSE = 1.78 days), and increased conditional R², indicating that variation in temperature response among stream reaches was substantial and better captured as a random effect.

We then tested whether the quadratic temperature term remained necessary when random slopes were included. The full model with both random slopes and a quadratic temperature effect provided better support (ΔAIC = 20.4) than the linear version, suggesting that each component contributed complementary information. We retained this final model (m16_rs) as it provided strong predictive performance while acknowledging both general and site-specific patterns in temperature–spawn timing relationships.

4.5 Interactions

To assess whether fixed-effect interactions provide additional explanatory power beyond what is captured by random slopes, we tested two models: one with a temp_90 × stream interaction (m201) and one with a temp_90 × year interaction (m202). Both were compared to the baseline random slope model with quadratic temperature (m16_rs).

m201 <- lmer(yday ~ temp_90 * stream + I(temp_90^2) + year + (1 + temp_90 | COMID), data = df_mod, REML = FALSE)
m202 <- lmer(yday ~ temp_90 * year + I(temp_90^2) + stream + (1 + temp_90 | COMID), data = df_mod, REML = FALSE)

Table 16: Model selection for m201 vs. m202.

Model	df	AIC	delta_AIC	R2_marginal	R2_conditional	RMSE	ICC
m202	20	11568.63	0.000	0.620	0.994	1.359	0.985
m201	24	12948.51	1379.886	0.736	0.985	1.786	0.944
m16_rs	17	12965.97	1397.340	0.713	0.984	1.782	0.945

Table 17: Parameter estimates for interaction models.

	m16_rs			m201: temp_90 * stream			m202: temp_90 * year
Predictors	Estimates	CI	p	Estimates	CI	p	Estimates	CI	p
(Intercept)	235.12	231.18 – 239.05	<0.001	234.55	230.27 – 238.83	<0.001	232.86	226.46 – 239.25	<0.001
temp 90	13.85	13.02 – 14.68	<0.001	14.99	13.22 – 16.76	<0.001	18.84	17.87 – 19.80	<0.001
temp 90^2	-0.86	-1.16 – -0.56	<0.001	-1.12	-1.43 – -0.82	<0.001	1.53	1.23 – 1.83	<0.001
stream [Beaver]	17.37	10.90 – 23.84	<0.001	17.93	11.04 – 24.83	<0.001	21.60	11.35 – 31.85	<0.001
stream [Big]	-0.17	-5.11 – 4.78	0.947	-1.07	-6.48 – 4.33	0.697	-3.78	-11.78 – 4.23	0.355
stream [Camas]	2.29	-2.84 – 7.42	0.381	2.00	-3.59 – 7.59	0.483	1.82	-6.51 – 10.14	0.668
stream [Elk]	11.88	5.12 – 18.64	0.001	11.74	4.53 – 18.95	0.001	15.89	5.13 – 26.64	0.004
stream [Loon]	2.85	-2.17 – 7.87	0.266	3.43	-2.21 – 9.06	0.233	1.71	-6.48 – 9.89	0.683
stream [Marsh]	7.91	2.55 – 13.27	0.004	9.15	3.35 – 14.96	0.002	9.50	0.82 – 18.18	0.032
stream [Sulphur]	14.69	8.97 – 20.42	<0.001	15.37	9.26 – 21.49	<0.001	24.50	15.44 – 33.56	<0.001
year [2003]	-4.99	-5.22 – -4.76	<0.001	-4.96	-5.19 – -4.73	<0.001	-7.03	-7.23 – -6.83	<0.001
year [2004]	2.76	2.52 – 3.00	<0.001	2.78	2.53 – 3.02	<0.001	2.96	2.77 – 3.15	<0.001
year [2005]	3.68	3.38 – 3.98	<0.001	3.69	3.39 – 4.00	<0.001	3.75	3.51 – 3.98	<0.001
temp 90 × stream [Beaver]				-2.81	-5.85 – 0.23	0.070
temp 90 × stream [Big]				0.83	-1.40 – 3.06	0.466
temp 90 × stream [Camas]				0.99	-1.42 – 3.39	0.421
temp 90 × stream [Elk]				0.68	-2.30 – 3.66	0.656
temp 90 × stream [Loon]				-1.29	-3.87 – 1.28	0.325
temp 90 × stream [Marsh]				-3.96	-6.48 – -1.44	0.002
temp 90 × stream [Sulphur]				-5.05	-7.75 – -2.35	<0.001
temp 90 × year [2003]							-3.89	-4.08 – -3.70	<0.001
temp 90 × year [2004]							0.63	0.40 – 0.87	<0.001
temp 90 × year [2005]							-1.29	-1.65 – -0.93	<0.001
Random Effects
σ2	3.36			3.37			1.96
τ00	45.06 COMID			50.42 COMID			114.83 COMID
τ11	12.18 COMID.temp_90			6.52 COMID.temp_90			17.85 COMID.temp_90
ρ01	-0.23 COMID			-0.35 COMID			0.00 COMID
ICC	0.94			0.94			0.99
N	104 COMID			104 COMID			104 COMID
Observations	3016			3016			3016
Marginal R2 / Conditional R2	0.713 / 0.984			0.736 / 0.985			0.620 / 0.994

Figure 21: Predicted spawn timing.

Model m202 showed a large AIC improvement (ΔAIC ≈ -1397), but inspection of its predicted effects revealed implausible temperature relationships—specifically, an inverted quadratic curve inconsistent with biological expectations. This suggests overfitting or confounding in the interaction structure.

Model m201 produced biologically reasonable predictions and modest AIC improvement (ΔAIC ≈ -18), but most interaction terms were non-significant, and variance explained (R²) remained nearly unchanged. These results indicate that stream-specific variation in thermal sensitivity is already well-captured by the random slope structure.

Given the limited inferential gain and added complexity of the interaction models, we retained m16_rs (quadratic temperature with COMID-specific random slopes) as our final model.

5 Model interpretation

5.1 Final Model

The final model included a linear and quadratic effect of temperature (temp_90, I(temp_90^2)), additive fixed effects for stream and year, and a random intercept and slope for temperature by COMID:

mod_final <- lmer(yday ~ temp_90 + I(temp_90^2) + stream + year + (1 + temp_90 | COMID), 
  data = df_mod, REML = TRUE)
# mod_final <- lmer(yday ~ temp_90 + I(temp_90^2) + stream + year + mean_elevation + (1 + temp_90 | COMID), 
#   data = df_mod, REML = TRUE)
# mod_final <- lmer(yday ~ temp_90 + I(temp_90^2) + year  + stream * mean_elevation + (1 + temp_90|COMID), data = df_mod, REML = FALSE)

This model balances explanatory power, biological realism, and parsimony. It captures both general patterns in spawn timing (via fixed effects) and local deviations in temperature response (via random slopes), and will serve as the basis for diagnostics, prediction, and ecological interpretation.

5.2 Model summary and fit

Parameter estimates (Table 18) indicate a significant nonlinear effect of temperature on spawn timing, with the quadratic term (–0.85) confirming a concave-down relationship—i.e., spawn timing advances with increasing temperature, but levels off at high temperatures.

Most stream and year effects were significant, capturing expected spatiotemporal structure in spawn timing. Notably, Big, Camas, and Loon were not significantly different from the reference level (Bear Valley).

The random effects structure reveals substantial variation across COMIDs. The standard deviation for random intercepts (√τ₀₀ = 7.05) and random slopes for temp_90 (√τ₁₁ = 3.55) indicates strong spatial heterogeneity in both baseline timing and temperature sensitivity (Table 18). The intraclass correlation (ICC) was high (0.95), reflecting strong grouping structure at the COMID level (Table 19).

Model fit was excellent: the marginal R² (variance explained by fixed effects) was 0.698, and the conditional R² (fixed + random effects) was 0.985 (Table 19). This suggests that the majority of explanatory power is derived from spatially varying thermal responses captured by the random slopes, with additional structure provided by the fixed effects.

Table 18: Parameter estimates for mod_final.

Parameter	Coefficient	SE	CI	CI_low	CI_high	t	df_error	p	Effects	Group
(Intercept)	235.09	2.11	0.95	230.95	239.22	111.50	2999	0.00	fixed
temp_90	13.90	0.43	0.95	13.06	14.74	32.50	2999	0.00	fixed
I(temp_90^2)	-0.85	0.15	0.95	-1.15	-0.55	-5.54	2999	0.00	fixed
streamBeaver	17.41	3.46	0.95	10.62	24.19	5.03	2999	0.00	fixed
streamBig	-0.19	2.65	0.95	-5.37	5.00	-0.07	2999	0.94	fixed
streamCamas	2.30	2.75	0.95	-3.08	7.69	0.84	2999	0.40	fixed
streamElk	11.97	3.61	0.95	4.88	19.06	3.31	2999	0.00	fixed
streamLoon	2.80	2.68	0.95	-2.47	8.06	1.04	2999	0.30	fixed
streamMarsh	7.93	2.87	0.95	2.30	13.56	2.76	2999	0.01	fixed
streamSulphur	14.69	3.06	0.95	8.69	20.69	4.80	2999	0.00	fixed
year2003	-5.00	0.12	0.95	-5.23	-4.77	-42.79	2999	0.00	fixed
year2004	2.77	0.12	0.95	2.53	3.01	22.57	2999	0.00	fixed
year2005	3.68	0.15	0.95	3.38	3.98	24.00	2999	0.00	fixed
SD (Intercept)	7.05	NA	0.95	NA	NA	NA	NA	NA	random	COMID
SD (temp_90)	3.55	NA	0.95	NA	NA	NA	NA	NA	random	COMID
Cor (Intercept~temp_90)	-0.23	NA	0.95	NA	NA	NA	NA	NA	random	COMID
SD (Observations)	1.83	NA	0.95	NA	NA	NA	NA	NA	random	Residual

Table 19: Model performance for mod_final.

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
12948.76	12948.97	13050.96	0.9845502	0.6979933	0.9488428	1.780765	1.833613

5.3 Residual diagnostics

Model diagnostics for the final model were generally strong (Figure ??). The posterior predictive check shows excellent agreement between the observed and model-predicted distributions of spawn timing, with overlapping density curves and no major deviations.

Plots of linearity and homoscedasticity indicate acceptable model performance. While the residual vs. fitted plot shows a slight trend, it does not appear strong enough to invalidate model assumptions. The spread of residuals is approximately constant across fitted values, though there is minor funneling at the lower end, likely reflecting skew in early spawn dates.

Influential observations are limited. A few data points exceed standard influence thresholds (|standardized residual| > 2 and high leverage), but none are extreme enough to warrant removal, and their leverage is modest. The model appears robust to outliers.

Collinearity is low: variance inflation factors (VIFs) for all fixed effects are well below the conservative threshold of 5, suggesting little concern about multicollinearity.

The normal Q-Q plot of residuals shows slight right-skew and some heavy tails, but the distribution is reasonably close to normal. Similarly, random effect Q-Q plots for intercepts and slopes show mostly linear trends with slight deviation at the tails, indicating acceptable assumptions for the mixed-effects structure.

Overall, the model shows no violations of key assumptions and is suitable for inference and prediction.

Figure 22: Residual diagnostics for mod_final.

5.4 Predictions against original data

Model predictions closely matched observed spawn timing, with predicted values aligning well along the 1:1 line (Figure 23). This supports strong overall model fit.

Figure 23: Predicted vs. observed spawn timing for Chinook Salmon across all years and streams. The dashed line shows the 1:1 line. Points represent individual redd observations.

5.5 Population-level effects

5.5.1 Marginal means and contrasts of yday at each factor level

We estimated marginal mean of yday at each factor level, averaging over the random effects, to provide an overall estimate of the effect in the population (Figure 24).

Figure 24: Predicted mean spawn dates by stream (A), year (B) and stream and year (C), from the final mixed-effects model (yday ~ temp_90 + I(temp_90^2) + stream + year + (1 + temp_90 | COMID)). Black points with black lines (A, B), and colored points with horizontal lines (C) represent estimated marginal means and 95% confidence intervals. Boxplots in panels A and B show the distribution of observed redd counts by group, with individual data points in grey. The modeled predictions represent marginal means accounting for fixed effects and averaged over random effects.

Significant differences in spawn timing were observed between many stream pairs (Table 20), particularly involving Loon (later spawning) and Sulphur (earlier spawning). For example, fish in Loon spawned significantly later than in Bear Valley, Camas, and Elk, while Sulphur exhibited significantly earlier timing than all other streams except Elk. These patterns reflect spatial heterogeneity in temperature and elevation across streams that is not fully captured by fixed effects alone.

Table 20: Pairwise contrasts among stream effects on predicted spawn timing. Contrasts represent estimated differences in predicted spawn day of year between streams from the final model. Positive values indicate later predicted spawn timing in the first stream compared to the second. P-values are uncorrected.

Level1	Level2	Difference	SE	CI_low	CI_high	t	df	p
Beaver	Bear Valley	-1.43	3.47	-8.23	5.38	-0.41	2999	0.68
Big	Bear Valley	-2.66	2.66	-7.88	2.56	-1.00	2999	0.32
Camas	Bear Valley	0.63	2.75	-4.77	6.03	0.23	2999	0.82
Elk	Bear Valley	-6.60	3.62	-13.71	0.50	-1.82	2999	0.07
Loon	Bear Valley	8.90	2.68	3.64	14.15	3.32	2999	0.00
Marsh	Bear Valley	6.72	2.87	1.08	12.35	2.34	2999	0.02
Sulphur	Bear Valley	-9.47	3.16	-15.66	-3.28	-3.00	2999	0.00
Big	Beaver	-1.23	3.18	-7.46	4.99	-0.39	2999	0.70
Camas	Beaver	2.06	3.27	-4.35	8.46	0.63	2999	0.53
Elk	Beaver	-5.18	4.01	-13.04	2.69	-1.29	2999	0.20
Loon	Beaver	10.33	3.24	3.98	16.67	3.19	2999	0.00
Marsh	Beaver	8.14	3.41	1.46	14.82	2.39	2999	0.02
Sulphur	Beaver	-8.04	3.54	-14.98	-1.10	-2.27	2999	0.02
Camas	Big	3.29	2.40	-1.42	8.00	1.37	2999	0.17
Elk	Big	-3.94	3.35	-10.51	2.62	-1.18	2999	0.24
Loon	Big	11.56	2.35	6.95	16.16	4.92	2999	0.00
Marsh	Big	9.38	2.57	4.33	14.42	3.64	2999	0.00
Sulphur	Big	-6.81	2.78	-12.26	-1.36	-2.45	2999	0.01
Elk	Camas	-7.23	3.43	-13.97	-0.50	-2.11	2999	0.04
Loon	Camas	8.27	2.45	3.47	13.06	3.38	2999	0.00
Marsh	Camas	6.09	2.66	0.87	11.30	2.29	2999	0.02
Sulphur	Camas	-10.10	2.91	-15.80	-4.40	-3.47	2999	0.00
Loon	Elk	15.50	3.40	8.84	22.17	4.56	2999	0.00
Marsh	Elk	13.32	3.56	6.34	20.30	3.74	2999	0.00
Sulphur	Elk	-2.87	3.71	-10.14	4.41	-0.77	2999	0.44
Marsh	Loon	-2.18	2.57	-7.23	2.87	-0.85	2999	0.40
Sulphur	Loon	-18.37	2.91	-24.07	-12.66	-6.31	2999	0.00
Sulphur	Marsh	-16.19	3.10	-22.27	-10.10	-5.22	2999	0.00

There was a clear trend toward later spawning over the four-year period (Table 21). Spawning in 2005 occurred significantly later than in all previous years. Differences between 2002 and 2003 were not statistically significant, but later years (2004 and especially 2005) were associated with a progressive delay in mean spawn timing. This temporal shift likely reflects interannual variability in temperature and flow conditions.

Table 21: Pairwise contrasts among year effects on predicted spawn timing. Contrasts represent estimated differences in predicted spawn day of year between years from the final model. Positive values indicate later spawning in the first year compared to the second. P-values are uncorrected.

Level1	Level2	Difference	SE	CI_low	CI_high	t	df	p
2003	2002	0.97	0.57	-0.15	2.10	1.70	2999	0.09
2004	2002	2.35	0.65	1.07	3.62	3.61	2999	0.00
2005	2002	6.18	0.61	4.98	7.37	10.17	2999	0.00
2004	2003	1.37	0.61	0.18	2.56	2.26	2999	0.02
2005	2003	5.20	0.71	3.80	6.60	7.28	2999	0.00
2005	2004	3.83	0.87	2.11	5.54	4.38	2999	0.00

5.5.2 Estimating response vs. relation

To visualize the model’s predictions across the temperature gradient, we estimated the relationship between spawn timing and 90-day pre-spawn mean temperature (temp_90) for different groupings: overall, by stream, and by year. This approach allows us to assess both general trends and context-specific responses.

Stream-specific predictions (see plot suggestion below) show a consistent pattern: spawn timing increases nonlinearly with 90-day mean stream temperature, leveling off at high temperatures. This plateau is consistent with biological expectations, as spawning may be constrained by environmental or physiological thresholds. Stream-to-stream variation in predicted timing reflects both fixed stream effects and COMID-specific random intercepts and slopes.

Year-specific predictions similarly show consistent thermal responses across years, with modest offsets in average spawn timing due to year effects. These predictions further validate the model’s generalizability and temporal consistency.

In addition, a combined stream-by-year plot (e.g., facet grid) confirms that the final model captures heterogeneity in both space and time without overfitting. Lines track the raw data closely across groups, particularly in mid-range temperatures where most observations are concentrated.

Together, these plots indicate that the final model effectively captures both nonlinear temperature effects and spatial variation in thermal sensitivity, while maintaining interpretability and predictive strength.

When stratified by stream and year, predictions captured both the nonlinear temperature response and variation in baseline spawn timing across sites and years (Figure ??). The curvature was consistent across years, while intercept shifts reflected known spatial patterns (e.g., later spawning in warmer, downstream reaches). These results confirm that the model accurately describes both average and context-specific phenological responses to temperature.

Figure 25: Predicted relationship between spawn timing and 90-day pre-spawn mean temperature by stream and year. Lines represent model predictions from the final mixed-effects model. Colored points show observed redd timing, shaded ribbons represent 95% confidence intervals for predictions.

Figure 26: Predicted relationship between spawn timing and 90-day pre-spawn mean temperature by stream and year. Lines represent model predictions from the final mixed-effects model. Colored points show observed redd timing, shaded ribbons represent 95% confidence intervals for predictions.

5.6 Group-level effects (deviations from fixed effects)

Random intercepts and slopes varied considerably among COMIDs, reflecting spatial heterogeneity in both average spawn timing and thermal sensitivity (Figure 27A). We found considerable spread in intercepts, reflecting differences in average spawn timing between reaches. The random slopes for temp_90 likewise varied meaningfully across COMIDs, indicating that temperature–spawn timing relationships are not constant across space.

Sites with earlier average spawn timing (lower intercepts) generally exhibited stronger positive responses to temperature (higher slopes), while later-spawning sites tended to show weaker temperature effects, a pattern also evident in the weak negative correlation between intercepts and slopes (r = -0.2; Figure 27B). This indicates that reaches with earlier average spawn timing (i.e., negative intercepts) tend to exhibit stronger temperature sensitivity (i.e., steeper positive slopes), whereas later-spawning reaches show weaker responses to temperature. Biologically, this suggests that early-spawning populations are more phenologically plastic and adjust their timing more closely to thermal cues, while late-spawning populations may be constrained by other factors—such as photoperiod, migration fatigue, or compressed spawning windows—resulting in diminished thermal responsiveness. These findings highlight spatial heterogeneity in phenological flexibility, with potential implications for how different populations may respond to climate change.

Figure 27: (A) COMID-specific random parameter estimates for intercepts (left) and slopes (right). Points represent best linear unbiased predictions (BLUPs) from the final model, with horizontal bars indicating ±1.96 standard errors. (B) Correlation between random intercepts and slopes for 90-day temperature across COMIDs. Each point represents a stream reach (COMID).

When grouped by stream, Bear Valley and Big Creek exhibited early average spawn timing and high thermal sensitivity, while Sulphur and Marsh Creeks had later average timing with flatter temperature responses (Figure 28).

Figure 28: Boxplots of random intercepts and slopes for 90-day pre-spawn temperature by stream. Each box represents the distribution of best linear unbiased predictions (BLUPs) for a COMID’s random intercept (average spawn timing) or slope (thermal sensitivity). Bars are colored by stream.

However, when examining individual random effects for each COMID and stream, we observed considerable variation in both intercepts and slopes within streams (Figure 29). For example, Bear Valley and Big Creek had some of the earliest average spawn timings, but also exhibited a wide range of thermal sensitivities. In contrast, Sulphur Creek had later average spawn timing but also showed considerable variability in its response to temperature.

Figure 29: Random intercepts and slopes for 90-day pre-spawn temperature by stream. Each bar represents the best linear unbiased prediction (BLUP) for a COMID’s random intercept (average spawn timing) or slope (thermal sensitivity). Bars are colored by stream.

Variation in temperature–spawn timing relationships across stream reaches (COMIDs) as estimated from the final mixed-effects model is shown in Figure 30). While the overall relationship is positive and nonlinear, individual COMID slopes and intercepts vary considerably. Some reaches show steeper increases in spawn timing with temperature (i.e., stronger thermal sensitivity), while others are relatively flat, indicating a weaker or more buffered response. Grouping by stream shows that some streams (e.g., Bear Valley, Marsh) exhibit tightly clustered trajectories, while others (e.g., Camas, Big) show more divergence. This variation likely reflects fine-scale differences in local hydrology, geomorphology, or biological factors that influence how fish respond to thermal cues within stream systems. The consistency of the overall trend, despite local heterogeneity, supports the biological relevance of temperature in structuring spawn timing.

Figure 30: Predicted spawn timing by 90-day pre-spawn temperature and COMID. Each line represents the predicted spawn timing for a specific COMID, colored by stream. The black line and shaded ribbon represents the 95% confidence interval for the population-level predictions.

5.7 Elevation effects embedded in random structure

Although mean elevation was excluded as a fixed effect due to collinearity with temperature and inconsistent global directionality, examination of random effects against elevation reveals spatial structure that elevation helps explain (Figure X). Random intercepts (average spawn timing) showed positive relationships with elevation in some streams (e.g., Big Creek), where higher-elevation sites tended to have later average spawning relative to the population mean. In contrast, other streams (e.g., Bear Valley, Beaver) showed little or no elevation pattern.

Random slopes (thermal sensitivity to temperature) exhibited similarly idiosyncratic patterns. In some cases (e.g., Camas, Marsh), thermal sensitivity declined with elevation, suggesting that fish at higher elevations may respond less strongly to interannual temperature variability. In others, relationships were weak or even opposite in direction.

Taken together, these patterns indicate that elevation influences both average spawn timing and thermal sensitivity, but in ways that differ across streams. This heterogeneity justifies our decision to capture elevation-linked variation through COMID-level random effects rather than imposing a single fixed elevation term.

Figure 31: Relationship between mean_elevation and (A) random intercepts and (B) slopes for 90-day pre-spawn temperature, as well as mean_elevation and (C) spawn date (yday) and (D) temp_90 . Points and lines are colored by stream, and solid lines represent linear fits.

References

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